In this paper, we establish the convergence of a temporally semidiscrete
approximation of an abstract optimal control problem governed by a
fractional/normal evolution equation with a final observation. This
approximation uses the variational discretization concept [1]} and
uses a discontinuous Galerkin method to discretize the state equation in time.
The discontinuous Galerkin method is the famous \(\text{dG}(0)\)  scheme for the
normal evolution equation, and is equivalent to the well-known L1 scheme
[2]} with uniform temporal grids for the fractional evolution
equations.
The derived numerical analysis is applied to a Dirichlet boundary control
problem.
We note that there are many works (see
[3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}
and the references therein) devoted to the numerical analysis of the fractional
diffusion equations with rough initial value and source term, but, to our
knowledge, no numerical analysis is available for the fractional diffusion
equation with rough Dirichlet boundary value. This paper also fills in this gap.
